In many clinical trials designed to determine if two groups (treatment and control) differ relative to some continuous outcome measure, a fixed number of subjects are randomly assigned to the treatment and control groups respectively, but not all of the subjects enter the study at the same time. Periodically, during the study, the variable of interest is measured on each of the subjects in the study at a given time. In the examples motivating this investigation, if the treatment is more effective than the control, then the mean difference between the two groups increases with time. Statistical inferences which utilize this ordering information, i.e. that the mean differences are increasing, will be studied. Because the response is measured for the same patient at different times, the data are not independent. Thus, order restricted inferences will be considered under correlation structures appropriate for these longitudinal clinical trials. Both normal theory procedures and nonparametric techniques will be employed to [obtain estimates and tests of hypotheses.] Two-stage procedures, which allow for an intermediate analysis of the data at the midpoint of the study, have great appeal in these clinical trials. If there is strong evidence that the treatment is better than the control, then the experiment can be terminated early. To date little has been published concerning two-stage procedures for longitudinal studies which utilize ordering information. The distribution theory needed to implement nonparametric tests of homogeneity with an order restricted alternative will be developed for these longitudinal studies. Asymptotic results will be obtained and small sample critical values tabled. A two-stage permutation test will also be derived and these tests will be compared to those currently available for two-stage clinical trials. Evaluation of proposed treatments for kidney disease, diabetes, tooth decay, arthritis, complications resulting from AIDS, to name a few, all represent trials where the results of the proposed research could have a major impact.